Noncommutative Geometry and Cherednik Algebras

非交换几何和切里德尼克代数

• 批准号: 0555750
• 负责人: Karen Smith
• 金额: $ 34.46万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555750&HistoricalAwards=false
• 关键词: Noncommutative; Geometry; Cherednik; Algebras

This project concerns the theory and application of ``noncommutative projective geometry'' or the interaction of projective algebraic geometry with noncommutative algebra. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled and the motivating theme behind much of this proposal will be to understand noncommutative surfaces. A large class of these algebras can be classified in terms of the ``naive'' blow-ups developed in collaboration with Keeler and Rogalski. Although these blow-ups are constructed in a manner reminiscent of commutative blowups, and depend upon geometric data, their structure is quite unlike the classical objects. A major portion of the project will be to further understand these objects and to extend their applications. The other major theme of the project will be in applications of this general theory to specific classes of algebras. A particularly useful technique, here, is to ``complete'' the category of modules over a noncommutative algebra to those over a graded algebra and then to apply noncommutative projective geometry. This has, for example, been used to relate rational Cherednik algebras in type A to Hilbert schemes, and to Haiman's work on the n! conjecture. The project will continue this research to gain a deeper understanding of these important algebras and their relation to other areas of mathematics, for example to integrable systems and to the study of invariant eigendistributions on symmetric spaces.Algebraic geometry, which is one of the oldest areas of modern mathematics, has its origins in the study of polynomial equations; for example a plane curve is the set of solutions of a polynomial equation in two variables. This leads to a rich interplay between that geometric object and the (commutative) algebra of the associated polynomials. Noncommutative algebra, which is a much younger subject, also has its origins in the theory of equations, in this case matrix equations, and in recent years has become increasingly important in many areas of mathematics (for example the theory of differential equations) and physics (Heisenberg's uncertainty principle is a classic illustration, but more subtle non-commutativity occurs, for example, in string theory). It has become apparent in recent years that there are definite, though often rather subtle, geometric structures hidden in these noncommutative objects and the interplay between the two has led to a rich theory, actually several theories, in their own right. These are collectively called noncommutative geometry.

这个项目涉及“非交换射影几何”的理论和应用或射影代数几何与非交换代数的相互作用。粗略地说，并通过类比交换的情况下，分次模的范畴扭转上的非交换分次环的二次，分别立方增长应被认为是非交换模拟的投影曲线，分别表面。这种直觉导致了显着数量的非平凡的见解和结果在非交换代数。事实上，分类非交换曲线（和二次增长的非交换分次环）的问题可以被认为是解决了，这个提议背后的动机主题将是理解非交换曲面。这些代数中的一大类可以根据与Keeler和Rogalski合作开发的“天真”爆破来分类。虽然这些爆破是以一种让人想起交换爆破的方式构造的，并且依赖于几何数据，但它们的结构与经典对象完全不同。该项目的主要部分将是进一步了解这些对象，并扩展其应用程序。该项目的另一个主要主题是将这一一般理论应用于特定类别的代数。一个特别有用的技术，在这里，是“完成”的范畴的模在一个非交换代数的那些分次代数，然后应用非交换射影几何。这已经，例如，被用来与合理的Cherednik代数在A型希尔伯特计划，并海曼的工作的n！猜想该项目将继续进行这项研究，以便更深入地了解这些重要的代数及其与其他数学领域的关系，例如与可积系统的关系和与对称空间上不变本征分布的研究的关系。例如，平面曲线是二元多项式方程的解的集合。这导致了几何对象和相关多项式的（交换）代数之间丰富的相互作用。非对易代数是一门年轻得多的学科，也起源于方程理论，在这里是矩阵方程，近年来在数学（例如微分方程理论）和物理学的许多领域变得越来越重要（海森堡的测不准原理是一个经典的例子，但更微妙的非对易性出现在弦理论中）。近年来，我们已经清楚地看到，在这些非对易物体中隐藏着明确的、尽管常常相当微妙的几何结构，这两者之间的相互作用已经导致了一个丰富的理论，实际上是几个理论。这些被统称为非对易几何。